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Integral Calculator

Antiderivatives and definite integrals with each technique spelled out. Every result is differentiated back to check it, and the area is shaded on the graph.

∫\int
Reads as xsin⁡(x)x \sin\left(x\right)
Try:
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  • f(x)

How to use the integral calculator

Type the integrand the way you would on a graphing calculator: x^2 e^x, 1/(x^2-4), sin^2(x). Choose Indefinite for an antiderivative or Definite to add bounds (use pi, e or inf freely). Each line of the solution names the technique, so you can follow it and reproduce it on an exam.

Choosing a technique

  • Table integral? Powers, exe^x, 1/x1/x, sine and cosine integrate directly. See integration rules.
  • Function times its inner derivative? Use u-substitution: in ∫2xcos⁡(x2) dx\int 2x\cos(x^2)\,dx the 2x2x is the derivative of x2x^2.
  • Product of unlike functions? Use integration by parts, ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du, choosing uu by LIATE (Logs, Inverse trig, Algebraic, Trig, Exponential).
  • Polynomial over polynomial? Divide if needed, factor the bottom, then use partial fractions.
  • Powers of sine and cosine? Use power-reduction identities such as sin⁡2x=1−cos⁡2x2\sin^2 x = \tfrac{1-\cos 2x}{2}.

Why a definite integral is an area

Slice the region under the curve into thin rectangles, add their areas, and let the slices get infinitely thin. That limit of Riemann sums is the definite integral, and the Fundamental Theorem of Calculus says you can get it from any antiderivative: ∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a). See it build up step by step on what is an integral? Popular single results: integral of ln x, integral of 1/x, integral of sin²x.

Questions students ask

What is the difference between an indefinite and a definite integral?

An indefinite integral (antiderivative) is a family of functions F(x)+CF(x) + C whose derivative is f(x)f(x). A definite integral ∫abf(x) dx\int_a^b f(x)\,dx is a single number: the signed area between the curve and the x-axis from aa to bb.

Which integration techniques does it use?

The basic table (power, exponential, log, trig, inverse trig), linearity, u-substitution, integration by parts (choosing u by the LIATE order), trig identities, polynomial long division and partial fractions. Every answer is checked by differentiating it back.

Why is there a “+ C”?

Constants disappear when you differentiate, so infinitely many functions share the same derivative. The graph on integration rules lets you slide C and watch the whole family move.

What if it can’t find an antiderivative?

Some functions, such as ex2e^{x^2} or sin⁡xx\frac{\sin x}{x}, have no antiderivative made of elementary functions. For definite integrals CalcViz then gives a high-precision numerical value (Gauss–Legendre quadrature) instead.

Can it do improper integrals?

Yes. Type inf or ∞ as a bound (for example 1 to inf for 1x2\frac{1}{x^2}). The value is found from the antiderivative’s limit and double-checked numerically.

What does negative area mean?

Area below the x-axis counts as negative in a definite integral. The graph shades it in a different colour. If you want total (unsigned) area, use the area under the curve calculator.